Soliton resolution and asymptotic stability of $N$-solitons to the Degasperis-Procesi equation on the line

TL;DR

This paper proves the soliton resolution conjecture and the asymptotic stability of N-soliton solutions for the Degasperis-Procesi equation, a model for shallow water waves, using advanced mathematical analysis of long-time behavior.

Contribution

It introduces a novel application of the $ar eta$-generalized Deift-Zhou method to analyze the long-time asymptotics of the DP equation, confirming soliton resolution.

Findings

Verification of the soliton resolution conjecture for the DP equation.

Demonstration of asymptotic stability of N-soliton solutions.

Derivation of different long-time asymptotic expansions in various space-time regions.

Abstract

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as a model delineating the propagation of shallow water waves, stands as a completely integrable system and admits a $3\times3$ matrix Lax pair. In this manuscript, we study the soliton resolution and large time behavior of solutions to the Cauchy problem of the DP equation with generic initial data in Schwarz space. Employing the $\bar \partial$-generalization of the Deift-Zhou nonlinear steepest descent method, we deduce different long time asymptotic expansions of the solution $u(x,t)$ in two distinct types of space-time regions. This result verifies the soliton resolution conjecture and asymptotic stability of $N$-soliton solutions for the DP equation.